Radiation thermodynamics with applications to lasing
and fluorescent cooling
Carl E. Mungan
Department of Physics, U.S. Naval Academy, Annapolis, Maryland 21402-5040
共Received 21 July 2004; accepted 29 October 2004兲
Laser cooling of bulk matter uses thermally assisted fluorescence to convert heat into light and can
be interpreted as an optically pumped laser running in reverse. Optical pumping in such devices
drives the level populations out of equilibrium. Nonthermal radiative energy transfers are thereby
central to the operation of both lasers and luminescent coolers. A thermodynamic treatment of their
limiting efficiencies requires a careful development of the entropy and effective temperatures of
radiation, valid for the entire range of light from the blackbody to the ideal laser limiting cases. In
particular, the distinct meaning and utility of the brightness and flux temperatures should be borne
in mind. Numerical examples help illustrate these concepts at a level suitable for undergraduate
physics majors. © 2005 American Association of Physics Teachers.
关DOI: 10.1119/1.1842732兴
I. INTRODUCTION
Energy transfers between thermodynamic systems are often labeled as either work or heat.1 However, it is generally
recognized that the exchange of electromagnetic radiation
cannot be neatly categorized as one or the other. Exceptions
are blackbody radiation, which is one of the three traditional
forms of heat transfer, and ideal laser radiation which can be
characterized as work by the Carathéodory definition.2 Some
authors explicitly distinguish radiation as a third category of
energy transfer.3 However there is nothing special about radiation in this regard; any irreversible transfer of energy between two systems generally involves a mixture of heat and
work. For example, when a block slides over a rough table,
there is both mechanical work involved in the bending of
asperities on the surfaces of the block and the table, and heat
transfer between portions of the block and table at different
temperatures.4
The primary reason for delineating heat from work is to
introduce the second law of thermodynamics and arrive at
the concept of entropy. The limiting efficiency of an engine
or refrigerator depends on the entropies of the input and output energy fluxes. This paper explicitly treats photon engines
and optical coolers, thereby requiring a robust understanding
of the thermodynamic terms that apply to radiation.
II. BASIC CONCEPTS OF LASER COOLING
OF BULK MATTER
Before jumping into the abstractions of radiation entropy
and temperature, consider the principles of operation of an
optical engine or refrigerator. Figure 1 is a simple block diagram of a laser cooler, presented in the same style that physics textbooks typically use to introduce refrigerators. The
left-hand arrow denotes an external source of energy E in . In
a conventional refrigerator this external source does electrical work, but in the present context it is a directed, coherent
共and possibly polarized, particularly if the active material is
axial兲 narrowband laser beam.
The refrigerator consists of an optically active medium
that resonantly absorbs most, if not all, of the input laser
radiation. 共The absorption length may be increased either by
shaping the material into the form of an optical fiber or by
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Am. J. Phys. 73 共4兲, April 2005
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placing it in an optical cavity tuned to the pump wavelength.兲
Subsequently the medium relaxes by spontaneous emission
at a variety of different possible energy-level transitions, so
that the output fluorescence 共which carries away energy E out)
is multidirectional 共probably isotropic兲, incoherent, unpolarized 共or at most, partially polarized兲, and broadened in bandwidth. 共This fluorescence must be absorbed by some external
heat sink which is thermally insulated from the refrigerator.
Care must be taken to minimize reabsorption of the exiting
fluorescence, for example by employing a fiber sample geometry that enables the fluorescence to readily escape out its
sides.兲 In ordinary cases of laser absorption, E out is less than
E in , and the difference appears as an increase in the internal
energy of the material, thereby enabling lasers to cut, weld,
or ablate substances. However, the crucial and surprising
property of certain materials irradiated with lasers of just the
right color is that they emit light of a higher frequency 共and
hence photon energy兲 than they absorb; this property is
known as anti-Stokes or thermally assisted fluorescence. If
anti-Stokes emission predominates over resonant and Stokes
processes, then heat Q is withdrawn on average from the
medium in each excitation–relaxation cycle.
The simplest possible level scheme of such an absorber is
depicted in Fig. 2. The material has only three energy levels,
where level 1 is the ground state. The pump laser frequency
␯ p is tuned to resonance with the 1–2 transition, so that
E in⫽h ␯ p 共times the number of photons absorbed兲, where h
is Planck’s constant. Suppose that ␯ p is large enough that
nonradiative relaxation from the excited states to the ground
state is much less likely than radiative decay. In other words,
the fluorescence quantum efficiency is essentially unity. According to the energy gap law,5 this requirement is satisfied if
␯ p Ⰷ ␯ acceptor , where ␯ acceptor is the frequency of any other
modes of the substance to which the excited states can
couple. For a solid material, h ␯ acceptor⬵kT 共where k is Boltzmann’s constant兲 could represent phonon energies, where kT
is a typical thermal energy at the refrigerator’s operating
temperature T. Alternatively, the energy acceptors might be
localized vibrational or rotational modes. If we assume that
the radiative time constant for relaxation from the excited to
the ground states is relatively long, as is typical of fluorescent materials, and that the spacing between levels 2 and 3 is
© 2005 American Association of Physics Teachers
315
Fig. 1. Block diagram of the input and output energy flows for a laser
cooler.
small 共compared to kT), so that rapid nonradiative exchanges occur between them, then a thermal 共Boltzmann兲
distribution of population between these two excited levels
will be established prior to relaxation to the ground state. In
that case, some fraction of the active species 共viz. ions, atoms, molecules, or electrons兲 will be thermally promoted
into level 3. 共The fraction depends on the relative degeneracies and energy spacing of levels 2 and 3.兲 Consequently, the
average fluorescence photon energy h¯␯ f will be larger than
the laser pump photon energy h ␯ p , with the differency representing thermal energy withdrawn from the system, Q
⫽h(¯␯ f ⫺ ␯ p ) per cycle.
This cooling energy Q may be used to chill an external
load. Typical proposed applications6 of optical refrigerators
are to cool infrared detectors in satellites 共because competing
Stirling-cycle compressors introduce undesirable vibrations,
and thermoelectric coolers are highly inefficient at the liquidnitrogen temperatures of interest兲 and to cool small, specialized electronic or high-temperature-superconductor devices
共which can justify the expense of designing an optical
cooler兲. In the laboratory, the cooling load often is simply the
absorbing sample itself. In that case, the cooling energy first
reduces the temperature of the material 共suspended in an
optical-access vacuum chamber兲 and eventually balances the
heat load on the sample from the surroundings. In a welldesigned system, this load arises primarily from ambient
thermal radiation, implying that all three inputs and outputs
in Fig. 1 are radiative. Optical heat shields 共sometimes called
hot mirrors兲 can be used to further reduce this heat leak. The
sides of the active sample are coated with a material that
Fig. 3. Thermodynamic diagram analogous to Fig. 1 describing a laser
pumped by an external light source.
transmits the fluorescence but reflects the ambient blackbody
light 共assuming that the fluorescence and the blackbody radiation, which peaks at 10 ␮m in the case of 300 K surroundings, are spectrally well separated from one another兲.
The current record temperature drop for an unshielded
solid sample suspended in vacuum starting from room
temperature7 is an impressive 65 °C. This drop was achieved
for optical pumping of the rare-earth ion Yb3⫹ doped into a
heavy-metal fluoride glass 共ZBLAN兲. The advantages of
Yb3⫹ over other species are threefold: It consists of only two
bands of energy levels, thus avoiding excited-state absorption of the pump or fluorescence radiation; the two bands are
each split into a number of closely spaced levels, thus permitting efficient absorption of thermal energy from the host;
and the energy gap between the two bands is large 共corresponding to a near-infrared wavelength of 1 ␮m兲, thereby
minimizing nonradiative decay.8 Although helpful, these
three factors are not required for successful cooling of rareearth ions, as evidenced by recent results9 for Tm3⫹ . Laser
cooling of gaseous carbon dioxide,10 organic laser dyes in an
alcohol solution,11,12 and gallium arsenide heterostructures13,14 also have been reported.
The alert reader may notice that these systems 共rare-earth
doped solids, CO2 , solvated organic dyes, and semiconductor heterostructures兲 also make efficient lasers. The properties required for a material to perform well as an optical
cooler 共efficient luminescence, large optical cross sections,
minimal competing decay channels兲 also are properties that
promote lasing. In fact, one need simply reverse every arrow
in Fig. 1 to obtain the optically pumped laser sketched in Fig.
3. The optical pump might now be a flashlamp, which, like
the fluorescence in Fig. 1, is broadband, incoherent, undirected, and unpolarized. Some fraction of this radiative energy is converted into laser light and the rest appears as
waste heat to be removed say by circulating cooling water.
The realization that good lasers can also make good optical coolers suggests other possible cooling candidates. Both
Nd3⫹ :YAG and ruby have been explored for this role,15,16
although for technical reasons these particular materials have
not lived up to their theoretical cooling potential.17
III. REVIEW OF RADIATION THERMODYNAMICS
Fig. 2. A possible energy-level scheme of a cooling material. The three
states could represent electronic levels of an atom, vibrational levels of a
molecule, or levels within the valence and conduction bands of a semiconductor.
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Am. J. Phys., Vol. 73, No. 4, April 2005
The rate at which entropy is carried by a steady, unpolarized beam of light 共not necessarily collimated兲 in vacuum
across a surface A is given by
Carl E. Mungan
316
Ṡ⫽2kc ⫺2
冕冕冕
⍀
A
⌬␯
关共 1⫹n 兲 ln共 1⫹n 兲
⫺n ln n 兴 ␯ 2 d ␯ cos ␪ d⍀ dA,
共1兲
as discussed in the Appendix, where c is the speed of light,
and ␪ and ␾ are the polar and azimuthal angles, respectively,
at which a photon is traveling into element d⍀
⫽sin ␪ d␪ d␾ of a solid angle relative to the normal to an
element dA of the surface. 共Note that A could be part or all
of an arbitrary cross section of the beam, or it could be a
portion of the real surface of a material emitting or absorbing
the light.兲 The radiation is distributed over a set of optical
modes with mean occupation number n that depends on the
photon frequency ␯, the direction of travel ␪ and ␾ of the
photon 共within a range of solid angle ⍀兲, and two position
coordinates of the photon on the surface A 共affording a considerable opportunity for confusion if spherical coordinates
are used to describe it兲. The frequency integration is over the
relevant spectral bandwidth ⌬␯ of the light, where the factor
of ␯ 2 comes from the density of states. Finally the term in
square brackets in Eq. 共1兲 arises from Bose–Einstein statistics. It is crucial to realize that Eq. 共1兲 is valid whether or not
the radiation is thermal, that is, even for a nonequilibrium
photon distribution such as the fluorescence emission of
Fig. 2.
Entropy leads to the concept of radiation temperature. Two
definitions of this quantity have been proposed,18 called the
brightness and flux temperatures. Because the energy of a
photon is h ␯ and the average number of such photons in a
particular beam mode is n, the replacement of k times the
quantity in square brackets in Eq. 共1兲 by nh ␯ gives the rate at
which energy is carried by the beam 共that is, its power兲,
Ė⫽2hc
⫺2
冕冕冕
⍀
A
⌬␯
n ␯ d ␯ cos ␪ d⍀ dA.
3
共2兲
From Eq. 共2兲 the spectral radiance 共often referred to as
brightness,19 although properly speaking that is a photometric not a radiometric term兲 L ␯ ⬅dĖ/(dA⬜ d ␯ d⍀), where
dA⬜ ⬅cos ␪dA, is seen to be
L ␯ ⫽2nh ␯ 3 /c 2 .
共3兲
Thus an experimental measurement of the spectral radiance
共using an angle-resolved absolute-intensity spectrometer兲 directly gives the photon occupation numbers n.
The brightness temperature T B of a beam of radiation is
defined as the temperature of a blackbody whose spectral
radiance integrated over the bandwidth of the radiation is
equal to the 共integrated兲 radiance of the beam,
冕
⌬␯
n␯3 d␯⫽
冕
⌬␯
␯3
d␯,
exp共 h ␯ /kT B 兲 ⫺1
共4兲
according to Planck’s law. Further insight into the meaning
of the brightness temperature can be obtained by examining
two special cases of radiation. First consider a blackbody
emitting isotropically into a hemisphere 共via a small hole in
a furnace for example兲. Its energy flux density 共or irradiance兲
I E ⬅dĖ/dA is20
I EBB⫽
317
2␲h
c2
冕
⬁
0
␯3
d ␯ ⫽ ␴ T B4 .
exp共 h ␯ /kT B 兲 ⫺1
Am. J. Phys., Vol. 73, No. 4, April 2005
共5兲
Fig. 4. The spectral radiance of a 5800 K 共solar surface temperature兲 blackbody and a Gaussian source with a peak at ␭ 0 ⫽1 ␮ m representing a LED
with a hemispherical irradiance of 1 W/mm2 and a 25 nm bandwidth (w
⫽10 nm). Within this bandwidth, the area under the Planck curve is equal
to the area under the Gaussian. We could therefore describe this LED as
being as bright as the sun.
Here ␴ ⫽2 ␲ 5 k 4 /(15c 2 h 3 )⫽5.67⫻10⫺8 W m⫺2 K⫺4 is the
Stefan–Boltzmann constant.
Likewise its entropy flux density I S ⬅dṠ/dA is18
I BB
S ⫽
2␲k
c2
冕再
⬁
0
h ␯ /kT B
⫺ln关 exp共 h ␯ /kT B 兲 ⫺1 兴
1⫺exp共 ⫺h ␯ /kT B 兲
4
⫻ ␯ 2 d ␯ ⫽ ␴ T B3 .
3
冎
共6兲
The thermodynamic definition of temperature is
1 ⳵S
⬅
T ⳵E
冏
共7兲
.
V
If we apply Eq. 共7兲 to the cavity radiation escaping from the
interior of a furnace of volume V, we find
T⫽
dI EBB
dI BB
S
⫽
4 ␴ T B3 dT B
4 ␴ T B2 dT B
⫽T B .
共8兲
In agreement with Eq. 共4兲, the brightness temperature of
blackbody radiation is equal to the absolute temperature of
the furnace emitting that radiation. Hence the irradiance of
blackbody radiation is only a function of the temperature of
its source, and it is in this sense that one describes blackbody
radiation as ‘‘pure’’ heat or ‘‘low-grade’’ energy.
Consider next the case of narrowband radiation centered at
peak frequency ␯ 0 , such as might be emitted by an LED or
a real laser. It follows from Eq. 共4兲 that its mean brightness
temperature is
T̄ B ⫽
h␯0
,
k ln共 1⫹1/n̄ 兲
共9兲
where n̄⫽c 2 L̄ ␯ /(2h ␯ 30 ) according to Eq. 共3兲. Here L̄ ␯ is the
spectral radiance of the source averaged over its bandwidth
⌬␯.
The intuitive meaning of Eq. 共9兲 is easier to grasp from a
graph such as Fig. 4. Suppose the source spectrum has a
Gaussian distribution
Carl E. Mungan
317
冋 冉 冊册
L ␭ ⫽L peak exp ⫺
1 ␭⫺␭ 0
2
w
2
共10兲
,
where L peak is the spectral radiance at the source’s peak
wavelength ␭ 0 ⫽c/ ␯ 0 and w is the spectral width corresponding to one standard deviation. If we define the source’s
bandwidth as
⌬␭⬅
冕
⬁
0
g 共 ␭ 兲 d␭,
共11兲
where g(␭)⬅L ␭ /L peak is the normalized profile, then one
can show that
⌬␭⫽w 冑2 ␲ ⬵2w 冑2 ln 2,
共12兲
which is the full-width at half-maximum of the source’s
spectrum. It is left as an exercise to verify that
L peak⬵L ␭BB共 ␭ 0 兲 ⫽
2hc 2 ␭ ⫺5
0
exp共 hc/kT̄ B ␭ 0 兲 ⫺1
.
I ENB⫽2 ␲ hc ⫺2 n̄ ␯ 30 ⌬ ␯ sin2 ␦ ,
共13兲
共14兲
and its entropy flux density is
⫺2
I NB
关共 1⫹n̄ 兲 ln共 1⫹n̄ 兲 ⫺n̄ ln n̄ 兴 ␯ 20 ⌬ ␯ sin2 ␦ .
S ⫽2 ␲ kc
共15兲
The analog of Eq. 共8兲 becomes
T⫽
dI NB
S
⫽
h ␯ 0 dn̄
⫽T̄ B
k ln共 1⫹1/n̄ 兲 dn̄
共16兲
using Eq. 共9兲 in the last step. Therefore the brightness temperature remains an absolute thermodynamic temperature for
nonequilibrium radiation.
It is worth noting what happens if either the bandwidth ⌬␯
or divergence ␦ of the source collapses to zero. If its energy
flux density is to remain finite, then Eq. 共14兲 implies that
n̄→⬁. In this limit, Eq. 共15兲 becomes
⫺2
I NB
关共 1⫹n̄ 兲 ln n̄⫺n̄ ln n̄ 兴 ␯ 20 ⌬ ␯ sin2 ␦
S ⬵2 ␲ kc
⫽
k NB ln n̄
I
→0.
h ␯ 0 E n̄
共17兲
That is, the entropy carried by monochromatic or unidirectional radiation is zero, so that one can characterize an ideal
laser beam as ‘‘pure’’ work or ‘‘high-grade’’ energy. Note
from Eq. 共16兲, however, that 兰 T dI S remains equal to the
318
T F⬅
Am. J. Phys., Vol. 73, No. 4, April 2005
IE
.
IS
共18兲
Equations 共14兲 and 共15兲 imply for a narrowband source that
T F⫽
n̄
h␯0
.
k 共 1⫹n̄ 兲 ln共 1⫹n̄ 兲 ⫺n̄ ln n̄
共19兲
If the source is no more than moderately bright, then n̄Ⰶ1,
and Eq. 共19兲 becomes
T F⬵
That is, to an excellent approximation one can determine the
brightness temperature of a narrowband source by plotting
its spectral radiance and then finding the blackbody curve
which passes through its peak, as in Fig. 4.
For a very bright source, Eq. 共9兲 becomes kT̄ B ⬵n̄h ␯ 0
⫽␭ 20 L̄ ␯ /2. For example, an unpolarized 1 mW red helium–
neon laser with a beam area of 1 mm2 , a divergence 共halfangle ␦兲 of 0.5 mrad corresponding to a solid angle of approximately ␲ ␦ 2 ⫽0.8 ␮ sr, and a bandwidth of 1 GHz has a
mean brightness temperature on the order of 2⫻1010 K. 19
共The detailed variation in the brightness temperature with
frequency and angle can easily be deduced for simple profiles such as Gaussians.21兲
If the narrowband radiation is independent of the angular
directions ␪ and ␾ over a circular cone of half-angle ␦, then
its energy flux density is
dI ENB
irradiance 共just as it did for blackbody radiation兲 and is not
zero, as one might have expected if this integral defined a
heat flux density. Entropy and heat are not directly related for
nonequilibrium distributions.
To analyze the efficiency of optical devices, it is useful to
introduce the flux temperature,
h␯0
h ␯ 0 /k
n̄
⫽
.
k 共 1 兲共 n̄ 兲 ⫺n̄ ln n̄ 1⫹ln共 1/n̄ 兲
共20兲
Equations 共9兲 and 共20兲 imply that T̄ B /T F →1 as n̄→0, so
that the flux and brightness temperatures are equal for dim,
narrowband radiation.
In contrast, T F ⫽0.75T B for blackbody radiation according
to Eqs. 共5兲 and 共6兲. At first glance this result is surprising,
because it implies that if a large body 共of surface area A) at
temperature T radiates away a small amount of heat Q into
free space, then the light carries away entropy S⫽4Q/3T
according to Eq. 共18兲, rather than Q/T. This entropy transport appears to disagree with the first equality in Eq. 共8兲,
which can be rewritten in this context as Ṡ⫽ 兰 dQ̇/T. But in
fact Eq. 共8兲 does not imply that S⫽ 兰 dQ/T⫽Q/T, because
Q̇ is only a function of temperature 共specifically dQ̇
⫽4 ␴ AT 3 dT), whereas Q is implicitly being interpreted as a
function of time at the 共nearly兲 constant temperature of the
emitter. The lesson is to exercise caution when relating the
fluxes to the total energy and entropy. The light carries away
entropy 4Q/3T, while the entropy of the body decreases by
Q/T, and thus the total entropy change of the universe is
positive in this irreversible emission process.
On the other hand, we expect entropy to balance if a hot
body at temperature T⫹⌬T is radiatively coupled to surroundings at temperature T that are only infinitesmally
smaller in temperature by ⌬T. In this case, the net rate of
radiative energy transport from the hot body to the surroundings is
Q̇⫽ ␴ A 关共 T⫹⌬T 兲 4 ⫺T 4 兴 ⬵4 ␴ AT 3 ⌬T,
共21兲
while the net rate of entropy transport is
4
Ṡ⫽ ␴ A 关共 T⫹⌬T 兲 3 ⫺T 3 兴 ⬵4 ␴ AT 2 ⌬T.
3
共22兲
This time S⫽Q/T, as one would predict from the quasiequilibrium of the bodies with the radiation.
Returning to our discussion of Eq. 共19兲, for a very bright
source (n̄Ⰷ1) we obtain
T F⬵
n̄
h␯0
h ␯ 0 n̄
⫽
.
k 共 1⫹n̄ 兲 ln n̄⫺n̄ ln n̄
k ln n̄
共23兲
For ideal laser radiation, n̄→⬁ and thus T F becomes infinite,
which is consistent with the fact that it carries zero entropy at
a finite irradiance. One 共but not the only兲 way to get zero
Carl E. Mungan
318
entropy transport is if the radiation occupies a single optical
mode, corresponding to a monochromatic plane wave, so
that the multiplicity of its macrostate is unity. Also note that
Eqs. 共9兲 and 共23兲 imply that T̄ B /T F ⬵ln n̄ for a bright narrowband source. That is, the brightness temperature diverges
even more rapidly than the flux temperature as n̄→⬁.
So far it has been implicitly assumed that the fluorescent
emission in Fig. 1 is into free space. However if the cooling
sample is bathed in thermal radiation from the surroundings
at ambient temperature T A , then one needs to account both
for the occupation number n of fluorescence photons and the
distribution n A of ambient photons.22 The net energy and
entropy fluxes stem from the difference between the radiation leaving the body 共fluorescence plus thermal radiation
which is reflected or transmitted, assuming for simplicity that
the sample has negligible thermal emissivity兲 and that incident on the body 共thermal radiation兲.23 Hence
Ė net⫽2hc ⫺2
冕
⫺2hc ⫺2
⫽2hc ⫺2
冕
共 n⫹n A 兲 ␯ 3 d ␯ cos ␪ d⍀ dA
冕
共24兲
so that one recovers Eq. 共2兲. In contrast
Ṡ net⫽2kc
⫺2
冕
关共 1⫹n⫹n A 兲 ln共 1⫹n⫹n A 兲
⫹n A ln n A 兴 ␯ 2 d ␯ cos ␪ d⍀ dA,
共25兲
which does not simplify to Eq. 共1兲. That is, the entropy carried by the net radiative emission from the refrigerator is not
independent of the ambient temperature in general. However
for practical optical refrigerators, nⰇn A , and Eq. 共1兲 is an
excellent approximation to the net entropy flux. It is only in
the limit of vanishing fluorescence radiance that the ambient
radiation matters. Specifically Eq. 共25兲 can then be approximated as
冕
关共 1⫹n⫹n A 兲 ln共 1⫹n A 兲 ⫺ 共 n⫹n A 兲 ln n A
⫺ 共 1⫹n A 兲 ln共 1⫹n A 兲 ⫹n A ln n A 兴 ␯ 2 d ␯ cos ␪ d⍀ dA
⫽2kc ⫺2
冕
n ln共 1⫹1/n A 兲 ␯ 2 d ␯ cos ␪ d⍀ dA
⫽Ė net /T A
共26兲
after substituting the Planck expression for n A in the last
step. Consequently the lower limit of the fluorescence flux
temperature Ė net /Ṡ net is the ambient temperature T A as the
pump power in Fig. 1 is reduced toward zero. Even if the
laser pump is shut off, there remains some weak fluorescence
stimulated by the absorption of ambient thermal radiation.
319
E out⫽E in⫹Q.
Am. J. Phys., Vol. 73, No. 4, April 2005
共27兲
The cooling coefficient of performance is defined in the
usual way for a refrigerator as
␩⬅
Q
.
E in
共28兲
The maximum value of ␩ is the Carnot limit, ␩ C , and is
determined by the second law of thermodynamics. The entropy carried away by the fluorescence cannot be less than
the sum of the entropy withdrawn from the cooling sample
and the entropy transported in by the pump laser,
共29兲
where T is the steady-state operating temperature of the refrigerator, and T f and T p are the flux temperatures of the
fluorescence and pump radiation, respectively. The reversible
Carnot limit is obtained by choosing the equality sign in Eq.
共29兲. If one substitutes Eqs. 共27兲 and 共28兲 into Eq. 共29兲, we
obtain
␩ C⫽
⫺ 共 n⫹n A 兲 ln共 n⫹n A 兲 ⫺ 共 1⫹n A 兲 ln共 1⫹n A 兲
Ṡ net⬵2kc ⫺2
The stage is set for a thermodynamic analysis of the efficiency with which the optical refrigerator in Fig. 1 converts
heat into light. According to the first law 共energy conservation兲,
E out Q E in
⭓ ⫹
,
Tf
T Tp
n A ␯ 3 d ␯ cos ␪ d⍀ dA
n ␯ 3 d ␯ cos ␪ d⍀ dA,
IV. LIMITING EFFICIENCIES OF OPTICAL
CONVERTERS
T⫺⌬T
,
T f ⫺T
共30兲
where ⌬T⬅TT f /T p . Equation 共30兲 can be interpreted as the
Carnot coefficient of performance of the engine-driven refrigerator depicted in Fig. 5. The engine extracts heat E in
engine
from a hot reservoir at temperature T p and rejects heat E out
to a cold reservoir at temperature T f . It thus produces work
W with a Carnot efficiency of
␩ engine⫽
T p ⫺T f
.
Tp
共31兲
This work then drives a refrigerator to draw heat Q out of a
cold reservoir at temperature T and to dump waste heat
fridge
E out
into a hot reservoir at temperature T f , with a Carnot
coefficient of performance given by
␩ fridge⫽
T
.
T f ⫺T
共32兲
Note that as the pump power Ė in is reduced, T f decreases—
with a lower limit of T A ⭓T according to Eq. 共26兲—and
hence ␩ fridge increases, but at the expense of an overall decreased cooling power ␩ CĖ in . The product of Eqs. 共31兲 and
共32兲 equals Eq. 共30兲, and the net input and output energy
fluxes of Fig. 5 reproduce those of Fig. 1 if one identifies
engine
fridge
⫹E out
. This scheme prompts the realization
E out⫽E out
that the engine is required only because the pump radiation is
not ‘‘pure work’’ for a real laser. If the pump laser were ideal,
then T p →⬁ so that ⌬T⫽0 and Eq. 共30兲 would directly reduce to Eq. 共32兲. In contrast, if one were to try to pump an
optical cooler with fluorescence radiation 共say by recycling
the light兲, then T p →T f and Eq. 共30兲 indicates that its coefCarl E. Mungan
319
Tf⫽
hc
1
,
k␭ 0 f 1⫹ln共 1/n̄ f 兲
共35兲
with
n̄ f ⫽
␭ 30 f
Ė f
.
2hc A sample⌬ ␯ f ␲
共36兲
The fluorescence spectrum has a peak wavelength of ␭ 0 f
⫽975 nm and a full-width at half-maximum of about ⌬␭ f
⫽35 nm⬵c ⫺1 ␭ 20 f ⌬ ␯ f at room temperature. The fluorescence is assumed to be emitted homogeneously and hemispherically from the surface of the cooling sample 共of area
A sample), which is taken to be a cylinder whose height and
diameter are each 3.0 cm, with a total power of Ė f ⫽40 W.
关Note that Ė p ⫽Ė f /(1⫹ ␩ ) according to Eqs. 共27兲 and 共28兲,
thus explaining the approximation in the specification of the
pump power.兴 Hence n̄ f ⫽6.4⫻10⫺4 and T f ⫽1760 K. 共An
exact calculation using a measured fluorescence spectrum17
gives a similar value of 1530 K, thereby justifying the approximations used here.兲 Clearly T p is so much larger than
T f that Eq. 共32兲 can be taken to be the Carnot coefficient of
performance of this optical refrigerator. Consequently ␩ C
⫽20% at room temperature, and it diminishes approximately
linearly to zero as T→0.
In contrast, the actual cooling coefficient of performance
is
¯ f 兲 /␭
¯f,
␩ actual⫽ 共 ¯␯ f ⫺ ␯ p 兲 / ␯ p ⫽ 共 ␭ p ⫺␭
Fig. 5. A heat engine operating between thermal reservoirs with a high
temperature of T p and a cold temperature of T f which is moderately larger
than ambient, whose output work W is used to drive a refrigerator, which
cools a load at temperature T somewhat less than ambient and rejects heat
into the reservoir at intermediate temperature T f .
ficient of performance would fall to zero. Thermodynamics
thus explains why optical cooling requires a laser pump for
efficient operation: The loss in entropy of the cooling load
must, at minimum, be compensated by the gain in entropy of
the radiation as it is converted from the input pump to the
output fluorescence. Therefore, it is not surprising that
Landau24 dismissed the practicality of fluorescent cooling in
1946.
Consider an example using actual values22 relevant to laser cooling of Yb3⫹ :ZBLAN. The laser pump is narrowband
and very bright, so that Eq. 共23兲 is applicable,
T p⫽
hc n̄ p
,
k␭ p ln n̄ p
共33兲
where
n̄ p ⫽
␭ 3p
Ė p
,
2hc ␲ R 2p ⌬ ␯ p ␲ ␦ 2p
共34兲
from Eq. 共3兲. Suppose that the pump laser has a power of
Ė p ⬇40 W, a beam radius of R p ⫽0.5 mm, a bandwidth of
⌬ ␯ p ⫽40 GHz, a divergence of ␦ p ⫽1 mrad, and a wavelength of ␭ p ⫽1030 nm. Then n̄ p ⬇109 , which justifies the
use of Eq. 共23兲, and T p ⬇7⫻1011 K. In contrast, the fluorescence is only moderately bright, so that Eq. 共20兲 is called for,
320
Am. J. Phys., Vol. 73, No. 4, April 2005
共37兲
according to the discussion in Sec. II, assuming that the
pump beam is entirely absorbed by the sample. In the case8
of Yb3⫹ : ZBLAN, ¯␭ f ⫽995 nm so that ␩ actual⫽3.5% at the
pump wavelength of 1030 nm used in Eqs. 共33兲 and 共34兲.
Although it might appear that one could increase the cooling
performance by tuning the pump laser to longer wavelengths,
in practice the measured value of ␩ is found to roll off because 共heat producing兲 trace impurity absorption begins to
dominate the rapidly decreasing ytterbium absorption. Furthermore, the cooling coefficient decreases faster than linearly as the operating temperature is decreased,22 because the
long-wavelength absorption coefficient decreases with decreasing temperature 共so that ␭ p has to be reduced兲, while
simultaneously the average fluorescence wavelength ¯␭ f increases 共due to diminishing phonon absorption within the
ground and excited bands of energy levels兲.
An alternative to pumping the refrigerator in Fig. 1 optically is to pump a semiconductor diode electrically.25 The
key to this electroluminescent cooling is to choose an applied
forward bias V which is sufficiently smaller than the average
bandgap emission energy h¯␯ divided by the electron charge
e. As in the case of laser cooling, one factor that determines
how much smaller the pump energy eV must be than h¯␯ to
achieve net cooling is the fluorescence quantum efficiency.
In particular, Auger processes and surface recombination
limit this efficiency. If we ignore such limitations, the Carnot
coefficient of performance is again given by Eq. 共32兲, because the electrical pump in this case is a source of ‘‘pure’’
work. In practice, however, the required electrical current
creates Joule heating of the diode, which is the primary reason that optical pumping is preferred.13 Unfortunately, the
large refractive index of semiconductors 共e.g., 3.6 for GaAs兲
leads to trapping of the fluorescence photons, which deCarl E. Mungan
320
creases the external fluorescence quantum efficiciency, as the
photons rattle around inside the sample and are thus more
likely to encounter a nonradiative decay channel. Berdahl26
has suggested that this total internal reflection could be frustrated by bringing an external absorber to within a few microns of the semiconductor surface. Another suggestion by
Berdahl is to reverse bias the diode so that it cools a neighboring sample by absorbing its thermal emission, a phenomenon known as negative luminescence.
The preceding thermodynamic analysis of photoluminescent cooling can be similarly applied to the optically pumped
laser in Fig. 3. We define its efficiency as
␩⬅
E out
.
E in
共38兲
An analysis similar to the derivation of Eq. 共30兲 can be used
to deduce a Carnot limit of
␩ C⫽
T p ⫺T
,
T p ⫺⌬T
共39兲
where ⌬T⬅TT p /T l . In this case, T is the steady-state operating temperature of the laser medium, and T l and T p are the
flux temperatures of the output laser and input pump radiation, respectively. As one might have guessed, ␩ C increases
as the ratio T p /T l of these two flux temperatures increases.27
In fact, the thermodynamic limit on the efficiency of the laser
becomes 100% as this ratio approaches unity, which is a big
advantage of diode-pumped laser systems. 共In practice, however, T p will always be less than T l because a set of mutually
incoherent diode laser beams is essentially being combined
into a single, high quality output beam.兲
The idea of pumping one laser with another suggests an
intriguing concept. One can connect the refrigerator of Fig. 1
to the laser of Fig. 3, and simultaneously pump both in such
a manner that all of the waste heat generated by the latter is
removed by the former. Remarkably, this scheme can be
implemented using the same active material for both
devices,28 by choosing the pump frequency ␯ p to be intermediate between the mean spontaneous emission frequency ¯␯ f
and the stimulated emission frequency ␯ l . In effect, one is
balancing the anti-Stokes cooling from the refrigerator
against the Stokes heating from the laser, resulting in what is
known as athermal or radiation-balanced lasing. A simultaneous optical and thermodynamic analysis of a model athermal laser based on an ytterbium-doped crystalline system
can be found in Ref. 29.
In particular, a statistical equation for radiation entropy
can be deduced from the Shannon information-theoretical
definition. The advantage of this approach over the traditional radiative heat transfer viewpoint is that it encompasses
nonequilibrium photon distributions. Once the entropy is defined, two effective temperatures of radiation can be introduced to parametrize it. A Carnot limit on the performance of
an optical engine or refrigerator is then easily derived from
the first and second laws of thermodynamics. As usual, this
limit would only be attainable for reversible operation of the
device. The calculations in this article indicate, for example,
that an ytterbium based solid-state laser cooler has a thermodynamic limit on its coefficient of performance of 20% for a
typical set of operating conditions. However, experimental
measurements to date have indicated that at best 3.5% of the
pump power has actually been converted into cooling using
this material. This discrepancy is indicative of the inefficiencies resulting from impurities in the samples and irreversibilities in the fluorescent emission. On the positive side, this
shortfall implies that the ultimate limits on optical cooling of
bulk matter have not been approached and that plenty of
room remains for new scientific and engineering developments in this emerging field of research.17
ACKNOWLEDGMENT
I thank the Research Corporation for its support of this
work.
APPENDIX: THE ENTROPY OF RADIATION
A recent derivation of Eq. 共1兲 can be found in Ref. 21, but
it is beyond the understanding of undergraduate students. A
more accessible treatment is in Ref. 18 and is outlined here.
Filling in the details in the derivation makes for a reasonable
student exercise. We begin from the Shannon definition of
the entropy,31
S⫽⫺k
321
Am. J. Phys., Vol. 73, No. 4, April 2005
共A1兲
P state ln P state ,
where the summation is over all possible states of the system, each of which has probability P state . In the present context, a particular state is defined by the occupation of N 1
photons in optical mode 1, N 2 photons in optical mode 2,
and so on. Now assume that the probability p i (N i ) of finding
N i photons in mode i is independent of the probability
p j (N j ) of finding N j photons in mode j whenever i⫽ j.
Then
V. CONCLUDING REMARKS
Laser cooling is now a well established laboratory technique with myriad practical applications. For atom cooling,
the concept frequently is explained from the point of view of
momentum, in which large numbers of photons are fired like
high-speed ping-pong balls against an oncoming bowlingball-like atom to translationally slow it down.30 Another explanation, appropriate both to the idea of the anti-Stokes
cooling of solids and Doppler cooling of atoms, is to use
energy conservation to balance the heat loss from the target
against the energy gain of the radiation. As good as these
explanations are, neither is able to explain why a laser pump
is required rather than say a collimated beam of narrowband
light from a high-intensity lamp. The simplest explanation of
this last issue invokes entropy as discussed in this paper.
兺
states
⬁
P state⫽
兿
i⫽1
共A2兲
p i共 N i 兲 ,
where
⬁
兺
N i ⫽0
p i 共 N i 兲 ⫽1
共A3兲
for any i.
Straightforward algebra leads to
⬁
S⫽⫺k
⬁
兺 兺
i⫽1 N i ⫽0
共A4兲
p i 共 N i 兲 ln p i 共 N i 兲 .
Next assume that the probability of finding one additional
photon in any mode is independent of the number already
occupying that mode. This assumption implies that
Carl E. Mungan
321
N
p i 共 N i 兲 ⬀q i i ,
共A5兲
where 0⭐q i ⬍1. If we normalize p i (N i ) in Eq. 共A5兲 according to Eq. 共A3兲, we obtain
N
p i 共 N i 兲 ⫽ 共 1⫺q i 兲 q i i .
共A6兲
But the definition of the mean occupation number is
⬁
n i⬅
兺
N i ⫽0
共A7兲
N i p i共 N i 兲 .
We substitute Eq. 共A6兲 into Eq. 共A7兲 and evaluate the sum to
obtain
n i⫽
qi
ni
⇒q i ⫽
.
1⫺q i
1⫹n i
共A8兲
Equation 共A8兲 is consistent with the fact that q i can be interpreted from Eq. 共A6兲 as either the relative probability of
there being one more photon in mode i, or the probability of
finding a nonzero number of photons in that mode. 共Consider
for example the cases where n i ⫽0, 1, or ⬁.兲 We next substitute Eq. 共A8兲 into Eq. 共A6兲 to obtain
Ni
p i共 N i 兲 ⫽
ni
共 1⫹n i 兲 N i ⫹1
,
共A9兲
and then substitute Eq. 共A9兲 into 共A4兲 to finally find
⬁
S⫽k
兺 关共 1⫹n i 兲 ln共 1⫹n i 兲 ⫺n i ln n i 兴 .
i⫽1
共A10兲
The density of photon states20 per polarization mode in a
frequency interval d ␯ and element of solid angle d⍀ is
c ⫺3 ␯ 2 d ␯ d⍀. We multiply this result by two because of the
two independent transverse directions of the polarization of
light, and by the speed c to obtain the entropy per unit time
rather than per unit distance. We also multiply by the element
of surface area dA, as well as by cos ␪ to project this area
element perpendicularly to the direction of photon propagation. We thereby convert the sum in Eq. 共A10兲 into the integral given by Eq. 共1兲.
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Carl E. Mungan
322